Intersection of two deterministic parity automata

Given two deterministic parity automata $$A_1=(Q_1,\Sigma,\delta,q_{01},c_1)$$ and $$A_2=(Q_2,\Sigma,\delta,q_{02},c_2)$$ with the finite set of states $$Q_i$$, the finite alphabet $$\Sigma_i$$, the transition function $$\delta_i : Q_i \times \Sigma \rightarrow Q_i$$, the initial state $$q_{0i}$$ in $$Q_i$$, and the coloring function $$c_i : Q_i \rightarrow \mathbb{N}$$, with $$i \in\{0,1\}$$.

How is the intersection of two deterministic parity automata formally defined? In particular, how are the coloring functions combined?

• Have you read topological extension of parity automata by Michał Skrzypczak? – Apass.Jack Jun 21 at 0:22
• I think that this is a good question, as I am not aware of any concise write-up of a construction to do so. In principle, one could treat the parity automata as deterministic Rabin automata, build the straight-forward intersection, and then apply the DRA->DPA construction from ArXiV/CoRR paper 1701.05738 that bases on index appearance records. But that is quite a detour, and a direction construction should be given in some textbook. – DCTLib Jun 21 at 14:21
• Yes, indeed that is my question, if there is a direct contruction of the intersection of two DPA in literature. Do you have any reference to the one for two deterministic rabin automaton? – kafka Jun 21 at 16:14